Debt and Cost of Capital
Kerry Back
Overview
- Adding debt increases enterprise value because of the tax savings, up to a point.
- Enterprise value is free cash flow discounted at the WACC.
- Adding debt does not change FCF.
- So adding debt must reduce the WACC.
- The effect is through taxes, not just because debt is cheaper capital.
Example
- All equity firm
- Issues perpetual risk-free debt with interest payment \(X\)
- Value of debt is \(D=X/r_f\).
- Tax savings per year is \(tX\), where \(t=\) tax rate.
- Government’s claim is reduced by \(tX/r_f = tD\).
- Enterprise value is increased by \(tD\).
If there were no taxes …
- Enterprise value \(EV\) is unchanged
- New equity value is \(E = EV-D\).
- All risk is now borne by \(E\) dollars of capital rather than \(EV\), so risk per dollar increases by \(EV/E\).
- Risk premium of equity should increase by \(EV/E\).
- New cost of equity is
\[r_{\text{new}} = r_f + (EV/E) (r_{\text{old}}-r_f)\]
- Without taxes, WACC is
\[\frac{E}{EV} \left[r_f + \frac{EV}{E}(r_{\text{old}}-r_f)\right] + \left(1-\frac{E}{EV}\right)r_f\]
- This is \(r_{\text{old}}\).
- Leverage wouldn’t change cost of capital if there were no taxes.
But there are taxes …
- With taxes, leverage still increases the cost of equity.
- With risk-free debt as before, all risk is borne by \(E\) dollars of capital rather than the old enterprise value.
- Old enterprise value is the new enterprise value \(EV\) minus \(tD\).
\[EV - tD = E + D - tD = E + (1-t)D\]
- So risk per dollar of capital rises by a factor of
\[\frac{E+(1-t)D}{E}= 1 + (1-t)\frac{D}{E}\]
- So equity risk rises less when there are taxes.
- The reason it rises less is that the IRS shares the risk increase.
- Because equity risk rises less, the WACC falls (straightforward algebra).
- Leverage reduces the cost of capital - up to a point.